We consider the asymptotic distribution of the bulk part first. It is a direct result of [1]} or [2]}. Therefore from Lemma REF  the limiting distribution of
\(\left( \sum _{j=1}^{p-M}f_{1}\left( \widetilde{{\bf \lambda }_{j}}\right)-(p-M)\int f_{1}(x)dF^{c_{nM},H_{2n}},\cdots ,\sum _{j=1}^{p-M}f_{h}\left( \widetilde{{\bf \lambda }_{j}}\right)-(p-M)\int f_{h}(x)dF^{c_{nM},H_{2n}} \right) \) 
